Prove that ⟨f, g⟩=^b[f(x) g(x)+f^'(x) g^'(x)] d x defines an inner product on the space C^1[a, b

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Prove that ⟨f, g⟩=^b[f(x) g(x)+f^'(x) g^'(x)] d x defines an...

Prove that $\langle f, g\rangle=\int_a^b\left[f(x) g(x)+f^{\prime}(x) g^{\prime}(x)\right] d x$ defines an inner product on the space $\mathrm{C}^1[a, b]$ of continuously differentiable functions on the interval $[a, b]$. Write out the corresponding norm, known as the Sobolev $\mathrm{H}^1$ norm; it and its generalizations play an extremely important role in advanced mathematical analysis, [49].

Key Concepts

Sobolev Spaces

Sobolev spaces are a class of function spaces that include information about a function's derivatives, not just the function itself. They are critical in the study of partial differential equations and variational problems because they allow functions that may not be classically differentiable to still have meaningful weak derivatives and well-defined norms, thereby facilitating advanced mathematical analysis in more general settings.

Continuously Differentiable Functions

The space of continuously differentiable functions, denoted C^1, consists of functions that are not only continuous but also have continuous first derivatives. This space provides a natural setting for analysis involving both function values and their rates of change, which is particularly useful when defining norms that incorporate derivatives, such as those found in Sobolev spaces.

Inner Product

An inner product is a function that assigns a real number to every pair of elements in a vector space, satisfying linearity, symmetry (or conjugate symmetry), and positive definiteness. It introduces the notion of angles and lengths in the space, allowing one to discuss geometric concepts and analytical properties in more abstract settings.

Norm Induced by an Inner Product

A norm provides a measure of the magnitude of elements in a vector space. When a norm is induced by an inner product, it is defined as the square root of the inner product of an element with itself. This relationship ties the geometric structure provided by the inner product to the analytical convenience of measuring size or distance in the space.

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